Question

Evaluate the expression. \({ }_{25} P_5\)

Short answer

The numerical value of the expression \({ }_{25} P_5\) is 127512000.

Step-by-step solution

Understand the notation

The expression \({ }_{25} P_5\) represents the number of ways to select and arrange 5 items from a set of 25, where the order of selection is significant. In clearer terms, this is a permutation of 25 items taken 5 at a time.

Apply the permutation formula

Apply the permutation formula to the given values. The permutation formula is given as \( P(n, r) = \frac{n!}{(n-r)!} \). Substituting \( n = 25 \) and \( r = 5 \) into the formula, the equation becomes \( P(25, 5) = \frac{25!}{(25-5)!} \).

Simplify the equation

Carry out the arithmetic within the equation to find the value of (25-5)! = 20!. Thus, the equation simplifies to \( P(25, 5) = \frac{25!}{20!} \). Generally, a factorial is the product of an integer and all the integers below it. In this case, there's no need to calculate the full factorials, as most of the terms cancel out. As a result, the equation can further simplify to \( P(25, 5) = 25 * 24 * 23 * 22 * 21 \).

Calculate the final value

Finally, multiply the remaining numbers together to obtain the value of the permutation. Thus, the numerical calculation becomes \( P(25, 5) = 127512000 \).

Key concepts

Factorial

In mathematics, a factorial of a number is the product of all positive integers less than or equal to that number. It's noted by an exclamation mark after the number, like this: 5!. For example, the factorial of 5 (written as 5!) is calculated as:

• 5 x 4 x 3 x 2 x 1 = 120

Factorials grow quite rapidly with larger numbers, which makes them crucial in permutations and combinations. They help us understand how many different ways items can be arranged. For instance, in the formula \( P(n, r) = \frac{n!}{(n-r)!} \), the factorial helps determine arrangements by eliminating repetitive counts, leading to a manageable solution. In permutations like \( { }_{25} P_5 \), the factorial reduces the complexity by canceling out terms, focusing only on the specific selection of interest.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations, permutations, and countable configurations. It provides a way to count possibilities and solve problems around arranging objects. Understanding both permutations and combinations is key here.

• Permutations consider the order in which items are arranged. For example, how many ways can you arrange 5 books on a shelf?
• Combinations do not consider order. Imagine choosing 3 out of 5 books to read without caring about the order.

In the problem given, \( { }_{25} P_5 \), it’s a permutations problem because the order in which we choose and arrange the 5 items from 25 is crucial. Combinatorics thus provides the foundational rules and formulas like the permutation formula used in the solution.

Arithmetic Sequences

Arithmetic sequences are sequences of numbers with a common difference between consecutive terms. Here, although the problem doesn't directly explore arithmetic sequences, understanding sequences and patterns helps to see how math problems like this one unfold.

• An arithmetic sequence: 2, 5, 8, 11, ... is achieved by adding 3 each step.
Number sequences and patterns provide a foundation for algorithms and mathematical solutions like the stepwise breakdown of permutations. When calculating a factorial or permutation as in the \( { }_{25} P_5 \), noticing patterns such as the decreasing numbers (e.g., 25, 24, 23, 22, 21) increases understanding of the multiplication series involved, even if not arithmetic by nature.